3.6 Network Equations

For a network with nb  buses, all constant impedance elements of the model are incorporated into a complex nb × nb  bus admittance matrix Ybus   that relates the complex nodal current injections I
 bus   to the complex node voltages V  :

Ibus = YbusV.

Similarly, for a network with nl  branches, the nl × nb  system branch admittance matrices Yf  and Yt  relate the bus voltages to the nl × 1  vectors If  and It  of branch currents at the from and to ends of all branches, respectively:


If [ ⋅ ]  is used to denote an operator that takes an n ×  1  vector and creates the corresponding n × n  diagonal matrix with the vector elements on the diagonal, these system admittance matrices can be formed as follows:


The current injections of (3.8)–(3.10) can be used to compute the corresponding complex power injections as functions of the complex bus voltages V  :


The nodal bus injections are then matched to the injections from loads and generators to form the AC nodal power balance equations, expressed as a function of the complex bus voltages and generator injections in complex matrix form as

gS(V,Sg) = Sbus(V ) + Sd − CgSg = 0.